The Business Secretary also announced an extra £84 million to ‘help accelerate their work.’
It comes as the UK announced a further 170 deaths of people with coronavirus - the lowest since the day after lockdown began.
Speaking at Sunday's Downing Street press conference, which was delayed due to technical difficulties, the cabinet minister said work by the University of Oxford to find an effective drug was "progressing well".
But while pointing out the government had now committed more than a quarter of a billion pounds towards developing a Covid-19 vaccine in the UK, Mr Sharma warned there was no certainty one would ever be found.
Alok Sharma said: "In order to definitively conquer this disease we need to find a safe, workable vaccine.
"Last month I announced a new vaccine task force to co-ordinate the efforts of Government, academia and industry in the critical mission to find a vaccine.
"I'm very proud of how quickly our scientists and researchers have come together in their efforts developing a vaccine that will combat coronavirus."
He added: "The first clinical trial of the Oxford vaccine is progressing well with all phase one participants having received their vaccine dose on schedule earlier this week.
"The speed at which Oxford University has designed and organised these complex trials is genuinely unprecedented."
The secretary of state also said Imperial College London was "making good progress" and would look to move into clinical trials for a vaccine by mid-June, with larger scale trials in October.
So far the Government had invested £47m in the Oxford and Imperial vaccine programmes, and announced a further £84m in new funding "to help accelerate their work", he said.
Mr Sharma added: "This new money will help mass-produce the Oxford vaccine so that if current trials are successful we have dosages to start vaccinating the UK population straight away."
He said the pharmaceutical giant AstraZeneca had finalised a "global licensing agreement" with Oxford University with government support, and added: "This means that if the vaccine is successful AstraZeneca will work to make 30 million doses available by September for the UK as part of an agreement for over 100 million doses in total."
He said "the UK will be first to get access" but that the government would also ensure that "we're able to make the vaccine available to developing countries at the lowest possible cost".
Mr Sharma also announced the UK's first vaccines manufacturing innovation centre is expected to open at the Harwell Campus in Oxfordshire in summer 2021, a year ahead of plan, after receiving an extra £93m in government cash.
He said: "The centre, which is already under construction, will have capacity to produce enough vaccine doses to serve the entire UK population in as little as six months."
But with no guarantee of a successful vaccine being found, he also stressed the need to look at other drug treatments for those who get the virus.
He reported that six drugs have entered initial live clinical trials, which could be expanded if the results were positive.